Decimal multiplication with powers of 10: Compute the exact value of 3 × 0.3 × 0.03 × 0.003 × 30. Show careful handling of decimal places and powers of 10 to avoid rounding errors.

Introduction / Context:This problem checks fluency with decimal multiplication and powers of 10. Multiplying several decimals often leads to misplaced decimal points. A structured approach using scientific notation prevents such mistakes and ensures an exact final answer.

Given Data / Assumptions:

• Expression: 3 × 0.3 × 0.03 × 0.003 × 30
• All quantities are exact decimals (no rounding required).
• Standard arithmetic precedence applies (only multiplication here).

Concept / Approach:Convert decimals to powers of 10 whenever helpful. Group convenient factors to simplify mental arithmetic. Track exponents carefully, because each decimal place corresponds to a negative power of 10 (for example, 0.3 = 3 × 10^-1).

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• .0000243 and .000243: Under-count decimal shift.
• .0243: Over-shifts the decimal by two places.

Common Pitfalls:

• Losing track of exponents when multiplying several decimals.
• Rounding mid-way instead of keeping exact values until the end.

Final Answer: